Problem 78
Question
Find a linear equation in slope-intercept form that models the given description. Describe what each variable in your model represents. Then use the model to make a prediction. In \(1995,\) the average temperature of Earth was \(57.7^{\circ} \mathrm{F}\) and has increased at a rate of \(0.01^{\circ} \mathrm{F}\) per year since then.
Step-by-Step Solution
Verified Answer
The linear model for this problem is given by \(y = 0.01x + 57.7\), where \(y\) represents the Earth's temperature and \(x\) corresponds to the years passed since 1995. Using this model, the predicted temperature for 2020 is \(57.95^{\circ}F\).
1Step 1: Formulating the Linear Equation
A linear equation in slope-intercept form is given by the equation \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. Applying this to our context, the rate of increase (0.01) serves as the slope (\(m\)) and the average temperature in 1995 (57.7) serves as the y-intercept (\(c\)). Therefore, the linear model in our case is \(y = 0.01x + 57.7\). In this equation, \(y\) represents the temperature of the Earth and \(x\) represents the years passed since 1995.
2Step 2: Making a Prediction
To make a prediction about the temperature in a given year, substitute the number of years passed since 1995 into the equation. For instance, to predict the temperature in 2020, subtract 1995 from 2020 giving 25 years. Thus, the predicted temperature is given by \(y = 0.01 * 25 + 57.7\).
3Step 3: Calculating the Predicted Temperature
Carrying out the calculation from the current equation yields \(y = 0.01 * 25 + 57.7 = 57.95\). Therefore, according to our model, the average temperature of the Earth in 2020 would be \(57.95^{\circ}F\).
Key Concepts
Slope-Intercept FormRate of ChangeTemperature PredictionMathematical Modeling
Slope-Intercept Form
The slope-intercept form is a way of writing linear equations. It's expressed as \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. This form is helpful because it visually represents how the dependent value \(y\) changes with respect to the independent value \(x\).
The slope \(m\) in the equation indicates the rate of change of \(y\) for each unit increase in \(x\). Meanwhile, the y-intercept \(c\) is where the line crosses the y-axis, representing the initial condition or starting point of \(y\) when \(x = 0\).
This structure makes it easy to graph a linear relationship and predict values, which is particularly useful in many real-world situations.
The slope \(m\) in the equation indicates the rate of change of \(y\) for each unit increase in \(x\). Meanwhile, the y-intercept \(c\) is where the line crosses the y-axis, representing the initial condition or starting point of \(y\) when \(x = 0\).
This structure makes it easy to graph a linear relationship and predict values, which is particularly useful in many real-world situations.
Rate of Change
The rate of change in a linear equation is essentially the slope, which shows how much the dependent variable changes with respect to the independent variable.
In the context of temperature changes over time, the rate of change tells us how fast or slow the temperature increases each year. This is crucial for understanding trends over time. For example, in the problem provided, the rate of change for Earth's temperature is given as \(0.01\degree F\) per year.
Understanding this rate is important for making predictions about future conditions, and helps highlight the impact of gradual changes over time.
In the context of temperature changes over time, the rate of change tells us how fast or slow the temperature increases each year. This is crucial for understanding trends over time. For example, in the problem provided, the rate of change for Earth's temperature is given as \(0.01\degree F\) per year.
Understanding this rate is important for making predictions about future conditions, and helps highlight the impact of gradual changes over time.
Temperature Prediction
Temperature prediction using linear models involves substituting values into a linear equation to project future temperatures. By using the slope-intercept equation derived from historical data, we can estimate future values.
In the exercise, the equation \(y = 0.01x + 57.7\) was used to model Earth's temperature from 1995 onward. To predict temperatures for any future year, you simply substitute \(x\) with the number of years since 1995. This provides a straightforward forecast based on consistent trends.
This kind of prediction is helpful for planning and responding to long-term changes, allowing for better preparedness and adaptation to environmental shifts.
In the exercise, the equation \(y = 0.01x + 57.7\) was used to model Earth's temperature from 1995 onward. To predict temperatures for any future year, you simply substitute \(x\) with the number of years since 1995. This provides a straightforward forecast based on consistent trends.
This kind of prediction is helpful for planning and responding to long-term changes, allowing for better preparedness and adaptation to environmental shifts.
Mathematical Modeling
Mathematical modeling refers to using mathematical expressions, like equations or systems of equations, to represent real-world phenomena. These models are vital for translating complex real-world situations into understandable and workable forms.
The linear equation \(y = 0.01x + 57.7\) serves as a mathematical model for Earth's temperature change over time starting from 1995. By modeling this data, we get a simplified representation of how temperatures increase, which can be used for analysis and future forecasting.
Such models help to simplify, visualize, and interpret data. They provide a basis for making informed predictions, decisions, and planning based on historical and current observations.
The linear equation \(y = 0.01x + 57.7\) serves as a mathematical model for Earth's temperature change over time starting from 1995. By modeling this data, we get a simplified representation of how temperatures increase, which can be used for analysis and future forecasting.
Such models help to simplify, visualize, and interpret data. They provide a basis for making informed predictions, decisions, and planning based on historical and current observations.
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